The present invention relates to a jitter measurement apparatus and a jitter measurement method.
A Time Interval Analyzer or an oscilloscope has conventionally been used in a period jitter measurement. Each of those methods is called a Zero-crossing Method. As shown in FIG. 1, for example, a clock signal (a signal under measurement) x(t) from a PLL (Phase-Locked Loop) under test 11 is supplied to a time interval analyzer 12. Regarding a signal under measurement x(t), a next rising edge following one rising edge fluctuates against the preceding rising edge as indicated by dotted lines. That is, a time interval Tp between two rising edges, namely a period fluctuates. In the Zero-crossing Method, a time interval between zero-crossings (period) is measured, a fluctuation of period is measured by a histogram analysis, and its histogram is displayed as shown in FIG. 2. A time interval analyzer is described in, for example, xe2x80x9cPhase Digitizing Sharpens Timing Measurementsxe2x80x9d by D. Chu, IEEE Spectrum, pp.28-32, 1988 and xe2x80x9cA method of Serial Data Jitter Analysis Using One-Shot Time Interval Measurementsxe2x80x9d by J. Wilstrup, Proceedings of IEEE International Test Conference, pp.819-823, 1998.
In addition, Tektronix Inc. and LeCroy Co. have recently been providing digital oscilloscopes each being able to measure a jitter using an interpolation method. In this jitter measurement method using the interpolation method (interpolation-based jitter measurement method), an interval between data having signal values close to a zero-crossing out of measured data of a sampled signal under measurement is interpolated to estimate a timing of zero-crossing. That is, a time interval between zero-crossings period) is estimated by a data-interpolation with a small error to measure a relative fluctuation of period.
That is, as shown in FIG. 3, a signal under measurement x(t) from the PLL under test 11 is inputted to a digital oscilloscope 14. In the digital oscilloscope 14, as shown in FIG. 4, the inputted signal under measurement x(t) is converted into a digital data sequence by an analog-digital converter 15. A data-interpolation is applied to an interval between data having signal values close to a zero-crossing in the digital data sequence by an interpolator 16. With respect to the data-interpolated digital data sequence, a time interval between zero-crossings is measured by a period estimator 17. A histogram of the measured values is displayed by a histogram estimator 18, and a root-mean-square value and a peak-to-peak value of fluctuations of the measured time intervals are obtained by an RMS and Peak-to-Peak Detector 19. For example, in the case in which a signal under measurement x(t) has a waveform shown in FIG. 5A, its period jitters are measured as shown in FIG. 5B.
On the contrary, we have proposed the xcex94xcfx86 method for measuring a jitter by obtaining a variable component (phase noise) of an instantaneous phase of a signal under measurement. This xcex94xcfx86 method is characterized in that an instantaneous phase of a signal under measurement is estimated using an analytic signal theory. FIG. 6 shows a processing block diagram of the xcex94xcfx86 method. An input signal is transformed into a complex analytic signal by a Hilbert pair generator 21. An instantaneous phase of an input signal is obtained from the complex analytic signal by an instantaneous phase estimator 22. A linear phase component is removed from the instantaneous phase by a linear trend remover 23 to extract a phase noise. With respect to this phase noise, a sample value closest to a zero-crossing point in a real part of the complex analytic signal is extracted by a zero-crossing resampler 24 to obtain a timing jitter sequence. A peak-to-peak value of the output of the zero-crossing resampler 24 is obtained by a xcex94xcfx86PP detector 25 as a peak-to-peak jitter xcex94xcfx86PP of the input signal. In addition, a root-mean-square value of the output of the zero-crossing resampler 24 is obtained by a xcex94xcfx86RMS detector 26 as a root-mean-square value xcex94xcfx86RMS of jitter of the input signal. Furthermore, a histogram of each sample value of the resampler 24 is displayed and estimated by a histogram estimator 27. This xcex94xcfx86 method is described in, for example, xe2x80x9cExtraction of Peak-to-Peak and RMS Sinusoidal Jitter Using an Analytic Signal Methodxe2x80x9d by T. J. Yamaguchi M. Soma, M. Ishida, T. Watanabe, and T. Ohmi, Proceedings of 18th IEEE VLSI Test Symposium, pp. 395-402, 2000.
In the jitter measurement method by the time interval analyzer method, a time interval between zero-crossings is measured. Therefore a correct measurement can be performed. However, since there is, in this jitter measurement method, a dead-time when no measurement can be performed after one period measurement, there is a problem that it takes a long time to acquire a number of data that are required for a histogram analysis. In addition, in an interpolation-based jitter measurement method in which a wide-band oscilloscope and an interpolation method are combined, there is a problem that a jitter histogram cannot correctly be estimated and a jitter is overestimated (overestimation). That is, there is no compatibility in measured jitter values between this jitter measurement method and the time interval analyzer method. For example, a measured result of jitter measured by a time interval analyzer for a clock signal of 400 MHz is shown in FIG. 7A, and a measured result of jitter measured by an interpolation-based jitter measurement method for the same clock signal is shown in FIG. 7B.
Those measured results are, a measured value by the time interval analyzer 7.72 ps (RMS) vs. a measured value by the interpolation-based jitter measurement method 8.47 ps (RMS), and the latter is larger, i.e., the measured value by interpolation-based jitter measurement method has overestimated the jitter value. In addition, the interpolation-based jitter measurement method cannot correctly estimate a Gaussian distribution having single peak.
It is an object of the present invention to provide a jitter measurement apparatus and its method that can estimate a jitter value having compatibility, similarly to the xcex94xcfx86 method, with a conventional time interval analyzer method, i.e., a correct jitter value in a shorter time period.
The jitter measurement apparatus according to the present invention comprises: analytic signal transformation means for transforming a signal under measurement into a complex analytic signal; instantaneous phase estimation means for obtaining an instantaneous phase of the signal under measurement from the complex analytic signal transformed by the analytic signal transformation means; zero-crossing timing estimation means for obtaining a zero-crossing timing sequence of the signal under measurement from the estimated instantaneous phase; period estimation means for obtaining an instantaneous period sequence of the signal under measurement from the zero-crossing timing sequence estimated by the zero-crossing timing estimation means; jitter detection means to which the instantaneous period sequence is inputted for obtaining a jitter of the signal under measurement.
In addition, it is desirable that the jitter measurement apparatus further comprises cycle-to-cycle period jitter estimation means to which the instantaneous period sequence is inputted for calculating its differential sequence and for outputting a cycle-to-cycle period jitter sequence to supply it to the jitter detection means.
In addition, it is desirable that the zero-crossing timing estimation means comprises: instantaneous phase data interpolation means to which the instantaneous phase data are supplied for interpolating instantaneous phase data between a plurality of instantaneous phase data around a predetermined phase value in the instantaneous phase data; zero-crossing data determination means for determining a data closest to the predetermined value in the data-interpolated instantaneous phase data, and timing estimation means for estimating a timing of the determined data.
In addition, it is desirable that, in the jitter measurement apparatus, the zero-crossing timing estimation means is means to which the instantaneous phase data are supplied for estimating a zero-crossing timing sequence by an inverse interpolation method from a plurality of instantaneous phase data around a predetermined phase value in the instantaneous phase data to output the zero-crossing timing sequence.
In addition, it is desirable that the analytic signal transformation means comprises: band-pass filtering means to which the signal under measurement is supplied for extracting only components around a fundamental frequency from the signal under measurement to limit the bandwidth of the signal under measurement; and Hilbert transformation means for Hilbert-transforming an output signal of the band-pass filtering means to generate a Hilbert pair of the input signal.
In addition, it is desirable that the analytic signal transformation means comprises: time domain to frequency domain transformation means to which the signal under measurement is supplied for transforming the signal under measurement into a both-sided spectrum signal in frequency domain; bandwidth limiting means for extracting only components around a positive fundamental frequency in the both-sided spectrum signal; and frequency domain to time domain transformation means for inverse-transforming an output of the bandwidth limiting means into a signal in time domain.
In addition, it is desirable that the analytic signal transformation means comprises: a buffer memory to which the signal under measurement is supplied for storing therein the signal under measurement; means for taking out the signal in the sequential order from the buffer memory such that the signal being taken out is partially overlapped with the signal taken out just before; means for multiplying each taken out partial signal by a window function; means for transforming each partial signal multiplied by the window function into a both-sided spectrum signal in frequency domain; bandwidth limiting means for extracting only components around a positive fundamental frequency of the signal under measurement from the both-sided spectrum signal transformed in frequency domain; means for inverse-transforming an output of the bandwidth limiting means into a signal in time domain; and means for multiplying the signal transformed in time domain by an inverse number of the window function to obtain a band-limited analytic signal.
In addition, it is desirable that the jitter measurement apparatus further comprises AD conversion means to which the signal under measurement is inputted for digitizing and converting an analog signal into a digital signal to input the digital signal to the analytic signal transformation means.
In addition, it is desirable that the jitter measurement apparatus further comprises waveform clipping means to which the signal under measurement is inputted for removing amplitude modulation components of the signal under measurement and for extracting only phase modulation components of the signal under measurement to supply a signal without amplitude modulation components to the analytic signal transformation means.
In addition, it is desirable that the jitter detection means is peak-to-peak detection means for obtaining a difference between the maximum value and the minimum value of a supplied sequence.
In addition, it is desirable that the jitter detection means is RMS detection means for obtaining a root mean square value (RMS value) of a supplied sequence.
In addition, it is desirable that the jitter detection means is histogram estimation means for obtaining a histogram of a supplied sequence.
A jitter measurement method according to the present invention has the steps of: transforming a signal under measurement into a complex analytic signal; estimating an instantaneous phase of the signal under measurement from the complex analytic signal; estimating a zero-crossing timing sequence of the signal under measurement from the instantaneous phase; estimating an instantaneous period sequence of the signal under measurement from the zero-crossing timing sequence; and obtaining a jitter of the signal under measurement from the instantaneous period sequence.
In addition, it is desirable that the jitter measurement method has the step of providing the instantaneous period sequence as an input for calculating its differential sequence and for outputting a cycle-to-cycle period jitter sequence of the signal under measurement.
In addition, it is desirable that the step of estimating a zero-crossing timing sequence comprises the steps of: interpolating instantaneous phase data between a plurality of instantaneous phase data around a predetermined phase value in the instantaneous phase data; determining a data closest to the predetermined phase value in the data-interpolated instantaneous phase data; and estimating a timing of the determined data.
It is desirable that a polynomial interpolation method is used in the step of interpolating.
In addition, it is desirable that a cubic spline interpolation method is used in the step of interpolating to interpolate the waveform data around a zero-crossing.
In addition, it is desirable that the step of estimating a zero-crossing timing sequence is a step of estimating a zero-crossing timing sequence by an inverse interpolation method from a plurality of instantaneous phase data around predetermined phase values in the instantaneous phase data. It is desirable that the inverse interpolation method is an inverse linear interpolation method.
In addition, it is desirable that the step of transforming the signal under measurement into an analytic signal comprises the steps of: extracting only components around a fundamental frequency from the signal under measurement to limit the bandwidth of the signal under measurement; and Hilbert-transforming the band-limited signal to generate a Hilbert pair of the input signal.
In addition, it is desirable that the step of transforming the signal under measurement into an analytic signal comprises the steps of transforming the signal under measurement into a both-sided spectrum signal in frequency domain; extracting only components around a positive fundamental frequency in the both-sided spectrum signal; and inverse-transforming the extracted components around the fundamental frequency into a signal in time domain.
In addition, it is desirable that the step of transforming the signal under measurement into an analytic signal comprises the steps of storing the signal under measurement in a buffer memory; taking out the signal in the sequential order from the buffer memory such that the signal being taken out is partially overlapped with the signal taken out just before; multiplying each taken out partial signal by a window function; transforming each partial signal multiplied by the window function into a both-sided spectrum signal in frequency domain; extracting only components around a positive fundamental frequency of the signal under measurement from the both-sided spectrum signal transformed in frequency domain; inverse-transforming the extracted spectrum signal having components around the fundamental frequency into a signal in time domain; and multiplying the signal transformed in time domain by an inverse number of the window function to obtain a band-limited analytic signal.
In addition, it is desirable that the jitter measurement method has a step of performing a waveform-clipping of the signal under measurement to remove amplitude components of the signal under measurement and to extract only phase modulation components of the signal under measurement and for transferring a signal without amplitude modulation components to the step of transforming the signal under measurement into an analytic signal.
In addition, it is desirable that the step of obtaining a jitter is a step of obtaining a difference between the maximum value and the minimum value of the instantaneous period sequence to calculate a peak-to-peak value.
In addition, it is desirable that the step of obtaining a jitter is a step of obtaining a standard deviation of the instantaneous period sequence to calculate an RMS value.
In addition, it is desirable that the step of obtaining a jitter is a step of obtaining a histogram data of the instantaneous period sequence.
In addition, it is desirable that the step of obtaining a jitter is a step of obtaining a difference between the maximum value and the minimum value of the cycle-to-cycle period jitter sequence to calculate a peak-to-peak value.
In addition, it is desirable that the step of obtaining a jitter is a step of obtaining a root mean square value of the cycle-to-cycle period jitter sequence to calculate an RMS value.
In addition, it is desirable that the step of obtaining a jitter is a step of obtaining a histogram data of the cycle-to-cycle period jitter sequence.
In addition, it is desirable that the step of obtaining a jitter is a step of obtaining a part or all of the peak-to-peak values, RMS values, and histogram data.
The principles of the present invention will be described below. In the description, a clock signal is used as a signal under measurement.
A jitter-free clock signal is a square wave having a fundamental frequency f0. This signal can be resolved by Fourier analysis into harmonics comprising frequencies f0, 3f0, 5f0, . . . Since a jitter corresponds to a fluctuation of a fundamental frequency of a signal under measurement, only signal components around the fundamental frequency are handled in a jitter analysis.
A fundamental sinusoidal wave component of a jittery clock signal (signal under measurement) can be expressed by an equation (1) when its amplitude value is A and its fundamental frequency is f0.
xe2x80x83A cos(xcfx86(t))=A cos(2xcfx80f0t+xcex8xe2x88x92xcex94xcfx86(t))xe2x80x83xe2x80x83(1)
That is, the xcfx86(t) is an instantaneous phase function of the signal under measurement, and can be expressed by a sum of a linear instantaneous phase component 2xcfx80f0t containing a fundamental frequency f0, an initial phase component xcex8 (can be assumed to be zero in a calculation), and a phase modulation component xcex94xcfx86(t).
A zero-crossing point of a signal under measurement (a timing when a fundamental sinusoidal wave signal crosses a zero level) can be obtained as a timing when an instantaneous phase xcfx86(t) becomes xcfx80/2+2nxcfx80 or 3xcfx80/2+2nxcfx80 (n=0, 1, 2, . . . ). Here, a timing point when an instantaneous phase xcfx86(t) becomes xcfx80/2+2nxcfx80 corresponds to a falling zero-crossing point of the signal under measurement, and a timing point when an instantaneous phase xcfx86(t) becomes 3xcfx80/2+2nxcfx80 corresponds to a rising zero-crossing point of the signal under measurement.
Therefore, by estimating an instantaneous phase xcfx86(t) of a signal under measurement, by obtaining a zero-crossing timing when the instantaneous phase xcfx86(t) becomes xcfx80/2+2nxcfx80 or 3xcfx80/2+2nxcfx80 (n=0, 1, 2, . . . ), and then by obtaining a time interval between two zero-crossings, an instantaneous period of the signal under measurement can be obtained. In addition, from the obtained instantaneous period, a period jitter corresponding to a period fluctuation and a cycle-to-cycle period jitter corresponding to a fluctuation between adjacent periods can be obtained. Moreover, by estimating a timing of a zero-crossing point using the interpolation method with a small error, the measurement errors of a timing jitter and a cycle-to-cycle period jitter can be reduced.
In the jitter measurement method according to the present invention, an instantaneous phase xcfx86(t) of a signal under measurement x(t) shown in FIG. 8 is first estimated. FIG. 9 shows a waveform of the estimated instantaneous phase xcfx86(t). Next a timing when the instantaneous phase xcfx86(t) of the signal under measurement becomes xcfx80/2+2nxcfx80 or 3xcfx80/2+2nxcfx80 (n=0, 1, 2, . . . ) is estimated using the interpolation method or the inverse interpolation method, and then a time interval (instantaneous period) T between two zero-crossings is measured. Here, the period is assumed to be n periods (n=0, 1, 2, . . . ). FIG. 10 shows an instantaneous period waveform T[n] measured when n=1. Finally an RMS value and a peak-to-peak value of period jitters are measured from the instantaneous period sequence T[n]. A period jitter J is a relative fluctuation of a period T against a fundamental period T0, and is expressed by an equation (2).
T=T0+Jxe2x80x83xe2x80x83(2)
Therefore, an RMS period jitter JRMS corresponds to a standard deviation of the instantaneous period T[n], and is given by an equation (3).
JRMS={square root over ( )}((1/N)xcexa3k=1N(T[k]xe2x88x92TM)2)xe2x80x83xe2x80x83(3)
In this case, N is the number of samples of the measured instantaneous period data, TM is a mean value of the instantaneous period data. In addition, a peak-to-peak period jitter JPP is a difference between the maximum value and the minimum value of the T[n], and is expressed by the equation (4).
JPP=maxk(T[k])xe2x88x92mink(T[k])xe2x80x83xe2x80x83(4)
FIG. 11 shows a histogram (FIG. 11C) of the period jitters measured by the jitter measurement method according to the present invention, a histogram (FIG. 11A) measured by the conventional time interval analyzer, and a histogram (FIG. 11B) measured by the xcex94xcfx86 method so that the histogram of the present invention can be compared with the histograms of the conventional time interval analyzer and the xcex94xcfx86 method. The 0 point on the lateral axis in each of FIGS. 11B and 11C corresponds to a period of approximately 2485 (ps), and FIGS. 11B and 11C show difference values obtained by the equation (2) based on the period of approximately 2485 (ps) as a reference value. It can be seen that the histogram of FIG. 11C looks like the histogram of FIG. 11A. In addition, FIG. 12 shows an RMS value and a peak-to-peak value of the period jitter measured by the jitter measurement method according to the present invention so that these values of the present invention can be compared with the corresponding values measured by the time interval analyzer and the xcex94xcfx86 method, respectively. Here, the peak-to-peak value JPP of the observed period jitter is substantially proportional to a square root of logarithm of the number of events (the number of zero-crossings). For example, in the case of approximately 5000 events, JPP=45 ps is a correct value. A JPP error in FIG. 12 is shown assuming that 45 ps is the correct value. As shown in FIG. 12, the measured values in the method of the present invention correspond to xe2x88x923.1% in JRM, and +1.0% in JPP of the respective values measured by the time analyzer, and it can be seen that respective differences are small. As shown in FIGS. 11A, 11B, 11C and 12, the jitter measurement method according to the present invention can obtain measured jitter values compatible with the measurement method by the conventional time interval analyzer by which correct measured values can be obtained.
Furthermore, the jitter measurement method according to the present invention can measure a cycle-to-cycle period jitter at the same time with a period jitter. A cycle-to-cycle period jitter JCC is a period fluctuation between contiguous cycles, and is expressed by an equation (5).
JCC[k]=T[k+1]xe2x88x92T[k]xe2x80x83xe2x80x83(5)
Therefore, by obtaining differentials between the instantaneous period data measured by the method of the present invention described above to calculate a root-mean-square and a difference between the maximum value and the minimum value of the instantaneous period data, an RMS value JCC,RMS and a peak-to-peak value JCC,PP of cycle-to-cycle period jitter can be obtained by equations (6) and (7), respectively.
JCC,RMS={square root over ( )}((1/M)xcexa3k=1MJ2CC[k])xe2x80x83xe2x80x83(6)
JCC,PP=maxk(JCC[k])xe2x88x92mink(JCC[k])xe2x80x83xe2x80x83(7)
In this case, M is the number of samples of the measured cycle-to-cycle period jitter data. FIGS. 13 and 14 show a waveform and a histogram of the obtained cycle-to-cycle period jitter JCC[n], respectively.
In the jitter measurement method according to the present invention, as described above, a tiring (zero-crossing point) when the instantaneous phase becomes xcfx80/2+2nxcfx80 or 3xcfx80/2+2nxcfx80 (n=0, 1, 2, . . . ) may be estimated to obtain an instantaneous period, or a timing when the instantaneous phase becomes another phase value may be estimated to obtain an instantaneous period.
In addition, in the jitter measurement method according to the present invention, a period jitter can also be estimated with high accuracy by removing, using waveform clipping means, amplitude modulation (AM) components of a signal under measurement to retain only phase modulation (PM) components corresponding to a jitter.
An analytic signal z(t) of a real signal x(t) is defined by a complex signal expressed by the following equation (8).
z(t)xe2x89xa1x(t)+jxxe2x80x2(t)xe2x80x83xe2x80x83(8)
In this case, j represents an imaginary unit, and an imaginary part xxe2x80x2(t) of the complex signal z(t) is a Hilbert transformation of a real part x(t).
On the other hand, Hilbert transformation of a time function x(t) is defined by the following equation (9).                                           x            xe2x80x2                    ⁡                      (            t            )                          =                              H            ⁡                          [                              x                ⁡                                  (                  t                  )                                            ]                                =                                    1              π                        ⁢                                          ∫                                  -                  ∞                                                  +                  ∞                                            ⁢                                                                    x                    ⁡                                          (                      τ                      )                                                                            t                    -                    τ                                                  ⁢                                  xe2x80x83                                ⁢                                  ⅆ                  τ                                                                                        (        9        )            
In this case, xxe2x80x2(t) is a convolution of the function x(t) and (1/xcfx80f). That is, Hilbert transformation is equivalent to an output at the time when the x(t) is passed through an entire-band-pass filter. However, the output xxe2x80x2(t) in this case has not been changed in terms of its magnitude of spectrum components, but its phase has been shifted by xcfx80/2.
Analytic signal and Hilbert transformation are described in, for example, xe2x80x9cProbability, Random Variables, and Stochastic Processesxe2x80x9d by A. Papoulis, 2nd edition, McGraw-Hill Book Company, 1984.
An instantaneous phase waveform xcfx86(t) of a real signal x(t) can be obtained from an analytic signal z(t) using the following equation (10).
xcfx86(t)=tanxe2x88x921[xxe2x80x2(t)/x(t)]xe2x80x83xe2x80x83(10)
Next, an algorithm for estimating an instantaneous phase using Hilbert transformation will be described. First, a signal under measurement x(t) shown in FIG. 15 is transformed into an analytic signal z(t) by applying Hilbert transformation to the signal under measurement x(t) to obtain a signal xxe2x80x2(t) corresponding to an imaginary part of the complex signal. FIG. 16 shows a transformed analytic signal. A real part x(t) is indicated by a solid line, and an imaginary part xxe2x80x2(t) is indicated by a dashed line. Here, a band-pass filtering process has been applied to the obtained analytic signal. This is because a jitter corresponds to a fluctuation of a fundamental frequency of a signal under measurement and hence only signal components around a fundamental frequency are handled in a jitter analysis. Next, an instantaneous phase function xcfx86(t) is estimated from the obtained analytic signal using the equation (10). Here, xcfx86(t) is expressed using principal values of phase in the range of xe2x88x92xcfx80 to +xcfx80, and has a discontinuity point at the proximity of a point where the phase changes from +xcfx80 to xe2x88x92xcfx80. FIG. 17 shows the estimated instantaneous phase function xcfx86(t). Finally, by unwrapping (that is, an integer multiple of 2xcfx80 is appropriately added to a principal value xcfx86(t)) the discontinuous instantaneous phase function xcfx86(t), a continuous instantaneous phase function xcfx86(t) from which discontinuity has been removed can be obtained. FIG. 18 shows an unwrapped continuous instantaneous phase function xcfx86(t).
A transformation from a real signal to an analytic signal can be realized by a digital signal processing using Fast Fourier Transformation.
First, FFT is applied to, for example, a digitized signal under measurement x(t) (400 MHz clock) shown in FIG. 19 to obtain a both-sided spectrum (having positive and negative frequencies) X(f). FIG. 20 shows the obtained both-sided spectrum X(f). Next, data around the fundamental frequency 400 MHz in the positive frequency components of the spectrum X(f) are retained, and the remaining data are made zero. In addition, the positive frequency components are doubled. These processes in frequency domain correspond to limiting bandwidth of the signal under measurement and transforming the signal under measurement into an analytic signal in time domain. FIG. 21 shows the obtained signal Z(f) in frequency domain. Finally, by applying inverse FFT to the obtained signal Z(f), a band-limited analytic signal z(t) can be obtained. FIG. 22 shows the band-limited analytic signal z(t).
Transformation to an analytic signal using FFT is described in, for example, xe2x80x9cRandom Data: Analysis and Measurement Procedurexe2x80x9d by J. S. Bendat and A. G. Piersol, 2nd edition, John Wiley and Sons, Inc., 1986.
When values of a function y=f(x) are given for discontinuous values x1, x2, x3, . . . , xn of a variable x, xe2x80x9cinterpolationxe2x80x9d is to estimate a value of f(x) for a value of x other than xk (k=1, 2, 3, . . . , n).
In the timing estimation using an interpolation method, for example as shown in FIG. 23, an interval between two measurement points xk and xk+1 that contains a predetermined value yc is interpolated in sufficient detail. After that an interpolated data closest to the predetermined value yc is searched, whereby a timing x when a function value y becomes the predetermined value yc is estimated. A timing estimation error is proportional to a time width that equally divides a time length between two measurement points xk and xk+1. That is, in order to decrease a timing estimation error, it is desirable that y=f(x) is interpolated by dividing a time length between the two measurement points xk and xk+1 into equal time widths and by making each time width as short as possible.
First, an interpolation method using a polynomial will be described. Polynomial interpolation is described in, for example, xe2x80x9cNumerical Analysisxe2x80x9d by L. W. Johnson and R. D. Riess, Massachusetts: Addison-Wesley, pp. 207-230, 1982.
When two points (x1, y1) and (x2, y2) on a plane are given, a line y=P1(x) that passes through these two points is given by an equation (11), and is unitarily determined.
y=P1(x)={(xxe2x88x92x2)/(x1xe2x88x92x2)}y1+{(xxe2x88x92x1)/(x2xe2x88x92x1)}y2xe2x80x83xe2x80x83(11)
Similarly, a quadratic curve y=P2(x) that passes through three points (x1, y1), (x2, y2) and (x3, y3) on a plane is given by an equation (12).                     y        =                                            P              2                        ⁡                          (              x              )                                =                                                                                          (                                          x                      -                                              x                        2                                                              )                                    ⁢                                      (                                          x                      -                                              x                        3                                                              )                                                                                        (                                                                  x                        1                                            -                                              x                        2                                                              )                                    ⁢                                      (                                                                  x                        1                                            -                                              x                        3                                                              )                                                              ⁢                              y                1                                      +                                                                                (                                          x                      -                                              x                        1                                                              )                                    ⁢                                      (                                          x                      -                                              x                        3                                                              )                                                                                        (                                                                  x                        2                                            -                                              x                        1                                                              )                                    ⁢                                      (                                                                  x                        2                                            -                                              x                        3                                                              )                                                              ⁢                              y                2                                      +                                                                                (                                          x                      -                                              x                        1                                                              )                                    ⁢                                      (                                          x                      -                                              x                        2                                                              )                                                                                        (                                                                  x                        3                                            -                                              x                        1                                                              )                                    ⁢                                      (                                                                  x                        3                                            -                                              x                        2                                                              )                                                              ⁢                              y                3                                                                        (        12        )            
In general, a curve of (N-1)th degree y=PNxe2x88x921(x) that passes through N points (x1, y1), (x2, y2) . . . (xN, yN) on a plane is unitarily determined, and is given by an equation (13) from the Lagrange""s classical formula.                     y        =                                            P                              N                -                1                                      ⁢                          (              x              )                                =                                                                                          (                                          x                      -                                              x                        2                                                              )                                    ⁢                                      (                                          x                      -                                              x                        3                                                              )                                    ⁢                                      xe2x80x83                                    ⁢                  …                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          x                      -                                              x                        N                                                              )                                                                                        (                                                                  x                        1                                            -                                              x                        2                                                              )                                    ⁢                                      (                                                                  x                        1                                            -                                              x                        3                                                              )                                    ⁢                                      xe2x80x83                                    ⁢                  …                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  x                        1                                            -                                              x                        N                                                              )                                                              ⁢                              y                1                                      +                                                                                (                                          x                      -                                              x                        1                                                              )                                    ⁢                                      (                                          x                      -                                              x                        3                                                              )                                    ⁢                                      xe2x80x83                                    ⁢                  …                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          x                      -                                              x                        N                                                              )                                                                                        (                                                                  x                        2                                            -                                              x                        1                                                              )                                    ⁢                                      (                                                                  x                        2                                            -                                              x                        3                                                              )                                    ⁢                                      xe2x80x83                                    ⁢                  …                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  x                        2                                            -                                              x                        N                                                              )                                                              ⁢                              y                2                                      +            …            +                                                                                (                                          x                      -                                              x                        1                                                              )                                    ⁢                                      (                                          x                      -                                              x                        2                                                              )                                    ⁢                                      xe2x80x83                                    ⁢                  …                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          x                      -                                              x                                                  N                          -                          1                                                                                      )                                                                                        (                                                                  x                        N                                            -                                              x                        1                                                              )                                    ⁢                                      (                                                                  x                        N                                            -                                              x                        2                                                              )                                    ⁢                                      xe2x80x83                                    ⁢                  …                  ⁢                                      xe2x80x83                                    ⁢                                      (                                                                  x                        N                                            -                                              x                                                  N                          -                          1                                                                                      )                                                              ⁢                              y                N                                                                        (        13        )            
In the interpolation by polynomial of degree (Nxe2x88x921), a value of y=f(x) for a desired x is estimated from N measurement points using the above equation (13). In order to obtain a better approximation of an interpolation curve PNxe2x88x921(x), it is desirable to select N points in the proximity of x.
Next cubic spline interpolation will be described. Cubic spline interpolation is described in, for example, xe2x80x9cNumerical Analysisxe2x80x9d by L. W. Johnson and R. D. Riess, Massachusetts: Addison-Wesley, pp. 237-248, 1982.
xe2x80x9cSplinexe2x80x9d means an adjustable ruler (thin elastic rod) used in drafting. When a spline is bended such that the spline passes through predetermined points on a plane, a smooth curve (spline curve) concatenating those points is obtained. This spline curve is a curve that passes through the predetermined points, and has the minimum value of square integral (proportional to the transformation energy of spline) of its curvature.
When two points (x1, y1) and (x2, y2) on a plane are given, a spline curve that passes through these two points is given by an equation (14).
y=Ay1+By2+Cy1xe2x80x3+Dy2xe2x80x3
Axe2x89xa1(x2xe2x88x92x)/(x2xe2x88x92x1)
Bxe2x89xa11xe2x88x92A=(xxe2x88x92x1)/(x2xe2x88x92x1)xe2x80x83xe2x80x83(14)
Cxe2x89xa1(⅙)(A3xe2x88x92A)(x2xe2x88x92x1)2
Dxe2x89xa1(⅙)(B3xe2x88x92B)(x2xe2x88x92x1)2
Here, y1xe2x80x3 and y2xe2x80x3 are the second derivative values of the function y=f(x) at (x1, y1) and (x2, y2), respectively.
In the cubic spline interpolation, a value of y=f(x) for a desired x is estimated from two measurement points and the second derivative values at the measurement points using the above equation (14). In order to obtain a better approximation of an interpolation curve, it is desirable to select two points in the proximity of x.
Inverse interpolation is a method of conjecturing, when a value of a function yk=f(xk) is given for a discontinuous value x1, x2, . . . , xn of a variable x, a value of g(y)=x for an arbitrary y other than discontinuous yk (k=1, 2, . . . , n) by defining an inverse function of y=f(x) to be x=g(y). In the inverse linear interpolation, a linear interpolation is used in order to conjecture a value of x for y.
When two points (x1, y1) and (x2, y2) on a plane are given, a linear line that passes through these two points is given by an equation (15).
y={(xxe2x88x92x2)/(x1xe2x88x92x2)}y1+{(xxe2x88x92x1)/(x2xe2x88x92x1)}y2xe2x80x83xe2x80x83(15)
An inverse function of the above equation is given by an equation (16), and a value of x for y can unitarily be obtained.
x={(yxe2x88x92y2)/(y1xe2x88x92y2)}x1+{(yxe2x88x92y1)/(y2xe2x88x92y1)}x2xe2x80x83xe2x80x83(16)
In the inverse linear interpolation, as shown in FIG. 24, a value of x=g(yc) for a desired yc is estimated from two measurement points (xk, yk) and (xk+1, yk+1) using the above equation (16), whereby a timing x for obtaining a predetermined value yc is unitarily be estimated. In order to reduce an estimation error, it is desirable to select two measurement points xk and xk+1 between which x is contained.
Waveform clipping means removes AM (amplitude modulation) components from an input signal, and retains only PM (phase modulation) components corresponding to a jitter. Waveform clipping is performed by applying the following processes to an analog input signal or a digital input signal; 1) multiplying the value of the signal by a constant, 2) replacing a signal value larger than a predetermined threshold 1 with the threshold 1, and 3) replacing a signal value smaller than a predetermined threshold 2 with the threshold 2. Here, it is assumed that the threshold 1 is larger than the threshold 2. FIG. 25A shows an example of a clock signal having AM components. Since the envelope of the time based waveform of this signal fluctuates, it is seen that this signal contains AM components. FIG. 25B shows a clock signal that is obtained by clipping the clock signal shown in FIG. 25A using clipping means. Since the time based waveform of this signal shows a constant envelope, it can be ascertained that the AM components have been removed.